Fluid Mechanics
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Appendix A

Quick reference

This section serves a quick reference for the key equations used in this book.

Gradient:

∇=∂∂xi+∂∂yj+∂∂zk\nabla = \frac{\partial}{\partial x} \mathbf{i} + \frac{\partial}{\partial y} \mathbf{j} + \frac{\partial}{\partial z} \mathbf{k}

Divergence:

∇⋅u=∂u∂x+∂v∂y+∂w∂z\nabla \cdot \mathbf{u} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}

Curl:

∇×u=(∂w∂y−∂v∂z)i+(∂u∂z−∂w∂x)j+(∂v∂x−∂u∂y)k\nabla \times \mathbf{u} = \left( \frac{\partial w}{\partial y} - \frac{\partial v}{\partial z} \right) \mathbf{i} + \left( \frac{\partial u}{\partial z} - \frac{\partial w}{\partial x} \right) \mathbf{j} + \left( \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \right) \mathbf{k}

Laplacian:

∇2=∂2∂x2+∂2∂y2+∂2∂z2\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}

Curl of a gradient:

∇×(∇T)=0\nabla \times (\nabla T) = 0

Divergence of a curl:

∇⋅(∇×u)=0\nabla \cdot (\nabla \times \mathbf{u}) = 0

Lagrangian derivative operator:

ddt=∂∂t+(u⋅∇)\frac{d}{dt} = \frac{\partial}{\partial t} + (\mathbf{u} \cdot \nabla)

Velocity as a gradient of a scalar potential:

u=∇ϕ\mathbf{u} = \nabla \phi

Continuity, Eulerian form:

∂ρ∂t+∇(ρu)=0\frac{\partial \rho}{\partial t} + \nabla (\rho \mathbf{u}) = 0

Continuity, Lagrangian form:

dρdt+ρ∇⋅u=0\frac{d\rho}{dt} + \rho \nabla \cdot \mathbf{u} = 0

Momentum, Cauchy:

∂u∂t+(u⋅∇)u=1ρ∇⋅σ+Fbρ\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} = \frac{1}{\rho} \nabla \cdot \boldsymbol{\sigma} + \frac{\mathbf{F}_b}{\rho}

Stress tensor as a combination of pressure and deviatoric stress:

σ=−pI+τ\boldsymbol{\sigma} = -p \mathbf{I} + \boldsymbol{\tau}

Momentum, Euler:

∂u∂t+(u⋅∇)u=−1ρ∇p\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} = - \frac{1}{\rho} \nabla p

Momentum, Navier-Stokes:

∂u∂t+(u⋅∇)u=−1ρ∇p+ν∇2u+Fbρ\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} = - \frac{1}{\rho} \nabla p + \nu \nabla^2 \mathbf{u} + \frac{\mathbf{F}_b}{\rho}

Momentum, with body force (gravity):

∂u∂t+(u⋅∇)u=−1ρ∇p+g+ν∇2u\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} = - \frac{1}{\rho} \nabla p + \mathbf{g} + \nu \nabla^2 \mathbf{u}

Momentum, Navier-Stokes, in scalar form:

∂u∂t+u∂u∂x+v∂u∂y+w∂u∂z=−1ρ∂p∂x+ν(∂2u∂x2+∂2u∂y2+∂2u∂z2)\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} = - \frac{1}{\rho} \frac{\partial p}{\partial x} + \nu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right)
∂v∂t+u∂v∂x+v∂v∂y+w∂v∂z=−1ρ∂p∂y+ν(∂2v∂x2+∂2v∂y2+∂2v∂z2)\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} = - \frac{1}{\rho} \frac{\partial p}{\partial y} + \nu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2} \right)
∂w∂t+u∂w∂x+v∂w∂y+w∂w∂z=−1ρ∂p∂z−g+ν(∂2w∂x2+∂2w∂y2+∂2w∂z2)\frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z} = - \frac{1}{\rho} \frac{\partial p}{\partial z} - g + \nu \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2} \right)

Equation of state, moist air:

p=ρRdT[1+q(RvRd−1)]p = \rho R_d T \left[1 + q \left(\frac{R_v}{R_d} - 1 \right) \right]

Equation of state, seawater:

ρ=ρ0[1−βT(T−T0)+βS(S−S0)−βp(p−p0)]\rho = \rho_0 \left[ 1 - \beta_T(T-T_0) + \beta_S(S-S_0) - \beta_p(p-p_0) \right]

Hydrostatic approximation:

dwdt=0\frac{dw}{dt} = 0
∂p∂z=−ρg\frac{\partial p}{\partial z} = -\rho g

Rate of change of a rotating vector:

(dCdt)I=Ω×C\left(\frac{d\mathbf{C}}{dt}\right)_I = \mathbf{\Omega} \times \mathbf{C}

Rate of change of a rotating vector in a rotating frame:

(dBdt)I=(dBdt)R+Ω×B\left(\frac{d\mathbf{B}}{dt}\right)_I = \left(\frac{d\mathbf{B}}{dt}\right)_R + \mathbf{\Omega} \times \mathbf{B}

Rate of change of velocity in a rotating frame:

(duRdt)R=(duIdt)I−2Ω×uR−Ω×(Ω×r)\left( \frac{d \mathbf{u}_R}{dt} \right)_R = \left( \frac{d \mathbf{u}_I}{dt} \right)_I - 2 \mathbf{\Omega} \times \mathbf{u}_R - \mathbf{\Omega} \times \left( \mathbf{\Omega} \times \mathbf{r} \right)

Navier-Stokes equation with rotation:

∂u∂t+(u⋅∇)u=−1ρ∇p−1ρ∇Φ−2Ω×u+ν∇2u\frac{\partial \mathbf{u}}{\partial t} + \left( \mathbf{u} \cdot \nabla \right) \mathbf{u} = - \frac{1}{\rho} \nabla p - \frac{1}{\rho} \nabla \Phi - 2 \mathbf{\Omega} \times \mathbf{u} + \nu \nabla^2 \mathbf{u}

Coriolis parameter:

f=2Ωsin⁡(θ)f = 2 \Omega \sin(\theta)

ff-plane approximation:

f=f0=2Ωsin⁡(θ0)f = f_0 = 2 \Omega \sin(\theta_0)

β\beta-plane approximation:

f=f0+βyf = f_0 + \beta y
β=∂f∂y=2Ωcos⁡(θ0)RE\beta = \frac{\partial f}{\partial y} = \frac{2\Omega\cos(\theta_0)}{R_E}

Geostrophic balance:

f×u=−1ρ∇p\mathbf{f} \times \mathbf{u} = - \frac{1}{\rho} \nabla p

Geostrophic velocity:

ug=−1ρf∂p∂yu_g = - \frac{1}{\rho f} \frac{\partial p}{\partial y}
vg=1ρf∂p∂xv_g = \frac{1}{\rho f} \frac{\partial p}{\partial x}

Rossby number:

Ro≡(u⋅∇)uf×u≈UfL\text{Ro} \equiv \frac{\left( \mathbf{u} \cdot \nabla \right) \mathbf{u}}{\mathbf{f} \times \mathbf{u}} \approx \frac{U}{fL}

Boussinesq approximation:

ρ=ρ0+δρ(x,y,z,t)\rho = \rho_0 + \delta \rho(x, y, z, t)
p=p0(z)+δp(x,y,z,t)p = p_0(z) + \delta p(x, y, z, t)

Boussinesq equations:

dudt+f×u=−1ρ0∇δp+bk\frac{d \mathbf{u}}{dt} + \mathbf{f} \times \mathbf{u} = - \frac{1}{\rho_0} \nabla \delta p + b \mathbf{k}
∇⋅u=0\nabla \cdot \mathbf{u} = 0
dTdt=T˙\frac{d T}{dt} = \dot{T}
dSdt=S˙\frac{d S}{dt} = \dot{S}
b=b(T,S,p)b = b(T, S, p)

Buoyancy:

b=−gδρρ0b = - g \frac{\delta \rho}{\rho_0}

Thermal wind balance:

∂ug∂z=−1f∂b∂y\frac{\partial u_g}{\partial z} = - \frac{1}{f} \frac{\partial b}{\partial y}
∂vg∂z=1f∂b∂x\frac{\partial v_g}{\partial z} = \frac{1}{f} \frac{\partial b}{\partial x}

Potential density:

ρθ=ρ+p0gzcs2\rho_\theta = \rho + \frac{p_0 g z}{c_s^2}

Brunt-Väisälä (buoyancy) frequency:

N2=−gρ~θ∂ρ~θ∂zN^2 = - \frac{g}{\widetilde{\rho}_\theta} \frac{\partial \widetilde{\rho}_\theta}{\partial z}

Static instability:

∂ρ~θ∂z<0(stable)\frac{\partial \widetilde{\rho}_\theta}{\partial z} < 0 \quad \text{(stable)}
∂ρ~θ∂z>0(unstable)\frac{\partial \widetilde{\rho}_\theta}{\partial z} > 0 \quad \text{(unstable)}

Shallow water momentum equation:

dudt+f×u=−g∇η\frac{d \mathbf{u}}{dt} + \mathbf{f} \times \mathbf{u} = - g \nabla \eta

Shallow water continuity equation:

∂η∂t+∇⋅(hu)=0\frac{\partial \eta}{\partial t} + \nabla \cdot (h \mathbf{u}) = 0

Inertial-gravity wave dispersion:

ω=f2+gH(k2+l2)\omega = \sqrt{f^2 + gH(k^2 + l^2)}

Gravity wave dispersion:

ω=gH(k2+l2)\omega = \sqrt{gH(k^2 + l^2)}

Inertial wave dispersion:

ω=f\omega = f

Kelvin wave:

u=u^0eyLdei(x−gHt)u = \widehat{u}_0 e^{\frac{y}{L_d}} e^{i(x - \sqrt{gH} t)}
η=Hgu^0eyLdei(x−gHt)\eta = \sqrt{\frac{H}{g}} \widehat{u}_0 e^{\frac{y}{L_d}} e^{i(x - \sqrt{gH} t)}

Rossby radius of deformation:

Ld=gHfL_d = \frac{\sqrt{gH}}{f}

Conservation of potential vorticity:

ddt(ζ+fh)=0\frac{d}{dt} \left( \frac{\zeta + f}{h} \right) = 0

Potential vorticity:

ζ+fh\frac{\zeta + f}{h}

Conservation of potential energy:

∂∂t(gh22)+∇(ugh22)+gh22∇⋅u=0\frac{\partial}{\partial t} \left( \frac{gh^2}{2} \right) + \nabla \left( \mathbf{u} \frac{gh^2}{2} \right) + \frac{gh^2}{2} \nabla \cdot \mathbf{u} = 0

Conservation of kinetic energy:

∂∂t(hu22)+∇⋅(uhu22)+gu∇(h22)=0\frac{\partial}{\partial t} \left( \frac{h \mathbf{u}^2}{2} \right) + \nabla \cdot \left( \mathbf{u} \frac{h \mathbf{u}^2}{2} \right) + g\mathbf{u}\nabla \left(\frac{h^2}{2}\right) = 0

Conservation of total energy:

∂E∂t=∂PE∂t+∂KE∂t\frac{\partial E}{\partial t} = \frac{\partial PE}{\partial t} + \frac{\partial KE}{\partial t}
∂∂t12(hu2+gh2)+∇⋅[u(12hu2+gh2)]=0\frac{\partial}{\partial t} \frac{1}{2} \left(h\mathbf{u}^2 + gh^2\right) + \nabla \cdot \left[ \mathbf{u} \left( \frac{1}{2} h\mathbf{u}^2 + gh^2\right) \right] = 0
∂E∂t+∇⋅(F)=0\frac{\partial E}{\partial t} + \nabla \cdot \left( \mathbf{F} \right) = 0

Energy flux:

F=u(12hu2+gh2)\mathbf{F} = \mathbf{u} \left( \frac{1}{2} h\mathbf{u}^2 + gh^2\right)

Rossby wave frequency:

ω=Uk−βk\omega = Uk - \frac{\beta}{k}

Rossby wave phase speed:

cp=U−βk2c_p = U - \frac{\beta}{k^2}

Rossby wave group speed:

cg=U+βk2c_g = U + \frac{\beta}{k^2}

Reynolds decomposition:

u=u‾+u′\mathbf{u} = \overline{\mathbf{u}} + \mathbf{u}'

Reynolds-averaged Navier-Stokes equation:

∂u‾∂t+∇⋅(u‾ u‾)=−1ρ∇p‾+ν∇2u‾+∇⋅(u′u′‾)\frac{\partial \overline{\mathbf{u}}}{\partial t} + \nabla \cdot \left( \overline{\mathbf{u}}\, \overline{\mathbf{u}} \right) = - \frac{1}{\rho} \nabla \overline{p} + \nu \nabla^2 \overline{\mathbf{u}} + \nabla \cdot \left( \overline{\mathbf{u}' \mathbf{u}'} \right)

Reynolds-averaged continuity equation:

∇⋅u‾=0\nabla \cdot \overline{\mathbf{u}} = 0

Turbulent Kinetic Energy:

k=12u′2‾k = \frac{1}{2} \overline{\mathbf{u}'^2}

Turbulent Kinetic Energy budget:

∂k∂t+u‾⋅∇k=−12∇⋅(u′u′u′‾)−(u′u′‾⋅∇)u‾−1ρu′∇p′‾+δρ′ρu′⋅g‾+ν∇2k−ν∇u′⋅∇u′‾\frac{\partial k}{\partial t} + \overline{\mathbf{u}} \cdot \nabla k = - \frac{1}{2} \nabla \cdot (\overline{\mathbf{u}' \mathbf{u}' \mathbf{u}'}) - (\overline{\mathbf{u}' \mathbf{u}'} \cdot \nabla) \overline{\mathbf{u}} - \frac{1}{\rho} \overline{\mathbf{u}' \nabla p'} + \overline{\frac{\delta \rho'}{\rho} \mathbf{u}' \cdot \mathbf{g}} + \nu \nabla^2 k - \nu \overline{\nabla \mathbf{u}' \cdot \nabla \mathbf{u}'}

Kolmogorov’s turbulence spectrum:

E(k)=Kε2/3(kε)5/3E(k) = \mathcal{K} \varepsilon^{2/3} \left( \frac{k}{\varepsilon} \right)^{5/3}

Wave phase:

ψ=kx−ωt\psi = kx - \omega t

Wave elevation:

η=acos⁡ψ\eta = a \cos\psi

Wave velocity potential:

ϕ=agωcosh⁡[k(z+h)]cosh⁡(kh)sin⁡ψ\phi = \frac{a g}{\omega} \frac{\cosh[k(z + h)]}{\cosh(kh)} \sin\psi

Wave orbital velocities:

u=∂ϕ∂x=−aωkcosh⁡[k(z+h)]cosh⁡(kh)cos⁡ψu = \frac{\partial \phi}{\partial x} = - \frac{a \omega}{k} \frac{\cosh[k(z + h)]}{\cosh(kh)} \cos\psi
w=∂ϕ∂z=aωksinh⁡[k(z+h)]cosh⁡(kh)sin⁡ψw = \frac{\partial \phi}{\partial z} = \frac{a \omega}{k} \frac{\sinh[k(z + h)]}{\cosh(kh)} \sin\psi

Wave particle displacements:

ζ=∫u dt=−aekzsin⁡ψ\zeta = \int u\ dt = - a e^{kz} \sin\psi
ξ=∫w dt=aekzcos⁡ψ\xi = \int w\ dt = a e^{kz} \cos\psi

Wave orbital accelerations:

ax=∂u∂t=aω2ekzsin⁡ψa_x = \frac{\partial u}{\partial t} = a \omega^2 e^{kz} \sin\psi
az=∂w∂t=−aω2ekzcos⁡ψa_z = \frac{\partial w}{\partial t} = - a \omega^2 e^{kz} \cos\psi

Linear gravity wave dispersion:

ω=gktanh⁡(kh)\omega = \sqrt{g k \tanh(kh)}

Deep water: kh→∞kh \to \infty

ω=gk\omega = \sqrt{g k}

Shallow water: kh→0kh \to 0

ω=ghk\omega = \sqrt{gh} k

Phase speed:

Cp=ωkC_p = \frac{\omega}{k}

Deep water: kh→∞kh \to \infty

Cp=gkC_p = \sqrt{\frac{g}{k}}

Shallow water: kh→0kh \to 0

Cp=ghC_p = \sqrt{gh}

Group speed:

Cg=∂ω∂kC_g = \frac{\partial \omega}{\partial k}

Deep water: kh→∞kh \to \infty

Cg=Cp2C_g = \frac{C_p}{2}

Shallow water: kh→0kh \to 0

Cg=CpC_g = C_p

Stokes drift (deep water):

uSt=a2ωke2kzu_{St} = a^2 \omega k e^{2kz}

Wave energy balance:

∂E∂t+∇⋅(CgE)=0\frac{\partial E}{\partial t} + \nabla \cdot \left( \mathbf{C_g} E \right) = 0